Question

If $a=sin\frac{\pi }{18}sin\frac{5\pi }{18}sin\frac{7\pi }{18}$, and $x$ is the solution of the equation. $y=2\left[x\right]+2$ and $y=3[x-2]$, where [x] denotes the integral part of x, then ‘a’ is equal to

• A. $\left[x\right]$
• B. $\frac{1}{\left[x\right]}$
• C. $2\left[x\right]$
• D. $[x{]}^2$

Answer 1

The correct option is B

$\frac{1}{\left[x\right]}$

$a=sin\frac{\pi }{18}sin\frac{5\pi }{18}sin\frac{7\pi }{18}$

$=sin{10}^{\circ }sin{50}^{\circ }sin{70}^{\circ }$

$=\frac{1}{2}\left[2sin{70}^{\circ }sin{10}^{\circ }\right]sin{50}^{\circ }$

$=\frac{1}{2}[cos{60}^{\circ }-cos{80}^{\circ }]sin{50}^{\circ }$

$=\frac{1}{4}sin{50}^{\circ }-\frac{1}{4}\left\{2cos{80}^{\circ }sin{50}^{\circ }\right\}$

$=\frac{1}{4}sin{50}^{\circ }-\frac{1}{4}(sin{130}^{\circ }-sin{30}^{\circ })$

$=\frac{1}{4}sin{50}^{\circ }-\frac{1}{4}sin{50}^{\circ }+\frac{1}{4}.\frac{1}{2}=\frac{1}{8}$

$y=2\left[x\right]+2$ and $y=3[x-2]$

$\Rightarrow 2\left[x\right]+2=3[x-2]=3\left[x\right]-6$

$\Rightarrow \left[x\right]=8$

$\therefore a=\frac{1}{\left[x\right]}$